Linear Shift-Invariant Systems
Introduction
A response of a system to an input signal can be described by a function \(\mathcal{H}\), which is called the impulse response of the system. The response of the system to an input signal \(f(x, y)\) is usually denoted by \(g(x, y)\), and is given by
\[\begin{align} g(x, y) = \mathcal{H}\{f(x, y)\} \end{align}\]Linear Shift-Invariant Systems
A system is said to be a linear shift-invariant system if it satisfies the following two properties:
\[\begin{align} \mathcal{H}\{\alpha f_1(x, y) + \beta f_2(x, y)\} &= \alpha \mathcal{H}\{f_1(x, y)\} + \beta \mathcal{H}\{f_2(x, y)\} &\text{for all } \alpha, \beta \in \mathbb{R}\\ \mathcal{H}\{f(x - x_0, y - y_0)\} &= g(x - x_0, y - y_0) &\text{for all } x_0, y_0 \in \mathbb{R} \end{align}\]Convolution with the Impulse Response
An two-dimensional impulse response \(\mathcal{H}\{\delta(x, y)\}\) can be used to describe the response of a linear shift-invariant system to an arbitrary input signal \(f(x, y)\).
\[\begin{align} f(x, y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(u, v) \delta(x - u, y - v) \, du \, dv \end{align}\]We can rewrite the above equation in the LSIS form as follows:
\[\begin{aligned} \mathcal{H}\{f(x, y)\} &= \mathcal{H}\left\{\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(u, v) \delta(x - u, y - v) \, du \, dv\right\}\\ &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \underbrace{f(u, v)}_{\text{constant}} \underbrace{\mathcal{H}\{\delta(x - u, y - v)\}}_{\text{impulse response}} \, du \, dv\\ \end{aligned}\]Thus, the response of a linear shift-invariant system to an arbitrary input signal \(f(x, y)\) can be described by the convolution of the input signal with the impulse response of the system. In other words, if the output of a impulse response \(\mathcal{H}\{\delta(x, y)\}\) is known, the output of the system to an arbitrary input signal \(f(x, y)\) can be obtained by convolving the input signal with the impulse response.
\[\begin{align} h(x, u, y, v) &= \mathcal{H}\{\delta(x-u, y-v)\} \\ g(x, y) &= \mathcal{H}\{f(x, y)\} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(u, v) h(x, u, y, v) \, du \, dv \end{align}\]where \(h(x, u, y, v)\) is the impulse response of the system to a two-dimensional impulse \(\delta(x-u, y-v)\). Since the system is LSI,
\[\begin{align} h(x, u, y, v) &= h(x - u, y - v)\\ g(x, y) &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(u, v) h(x - u, y - v) \, du \, dv \\ &= f(x, y)\star h(x, y) \end{align}\]